I am reading Dummit and Foote Ch 10: Modules and in particulat section 10.3 Generation of Modules, Direct Sums and Free Modules.

I am having problems understanding and interpreting Theorm 6 - see attached pdf for the two relevant pages of D&F.

Problems I have are as follows:

1. We are told to "identify A as a subset of F(A) by a

where

is the function which is 1 at a and zero elsehwhere"

Thus

F(A) - but how can A be a subset of F(A)? The members of F(A) are functions

,

,

, ... etc. The entries are functions not just simple points or elements of A. The nature of the elements of A and F(A) seem different, so can we "identify" the elements of A as a subset of F(A)?

Further to this point where is the function a

used as the proof progresses from this point?

2. Following the quote above regarding

, we read:

"We can, in this way, think of F(A) as all finite R-linear combinations of elements of A by identifying each function f with the sum

+

+ ..... +

where f takes the value

at

and is zero at all other elements of A. Moreover, each element of F(A) has a unique expression as such a formal sum."

So you can see what assumptions I am working on, I have sketched my understanding of functions f, g

F(A) and also my understanding of

- see pdf attached called sketches of functions

Given that D&F describe f as taking on the value

at

and being zero at all other elements, how can we show that each element of F(A) has a unique expression as a formal sum of the form

+

+ ..... +

.

[I am also somewhat confused by the fact that the functional values of f are

,

, etc and not

,

, etc ]

Can anyone help clarify this theorem for me?

Peter